Fig 1.1. I think that False Alarms cannot not be retrieved using the proposed method under SDT framework given that LRT doesn’t result in a decision rule X>C for unequal variance normal distributions.

Section 9.1. Many different procedures have been developed for hypothesis testing. In much scientific research, null-hypothesis testing is common, in which a specific null hypothesis (there is no effect) is contrasted with a general alternative hypothesis (there is some effect). The asymmetry of these hypotheses potheses introduces corresponding asymmetries in the conclusions (one can reject the null hypothesis but not the alternative). Signal-detection theory is closer to another hypothesis testing procedure, known as likelihood-ratio testing. In it, the hypotheses being tested are equally specific, and the test lets one make a decision in favor of one of them. The conclusions from such tests are symmetrical. I think that it might be imprecise to classify hypothesis testing in null-hypothesis testing and likelihood ratio testing. Likelihood ratio is a general method to find a test to decide between hypotheses independently on whether the hypotheses are symmetric or not. If the size \(\alpha\) happens to be fixed, then the hypothesis testing is called null-hypothesis testing.